ON THE NUMBER OF CYCLES OF GIVEN LENGTH OF A FREE WORD IN SEVERAL RANDOM PERMUTATIONS

Let w not-equal 1 be a free word in the symbols g1,...,g(k) and their inverses (i.e., an element of the free group F(k)). For any s1,...,s(k) in the group S(n) of all permutation of n objects, we denote by w(s1,...,s(k)) is-an-element-of S(n) the permutation obtained by replacing g1,...,g(k) with s1,...,s(k) in the expression of w. Let X(w,L)(n) (s1,...,s(k)) denote the number of cycles of length L of w(s1,...,s(k)). For fixed w and L, we show that X(w,L)(n), viewed as a random variable on S(n)k, has (for n --> infinity) a Poisson-type limit distribution, which can be computed precisely. (C) 1994 John Wiley & Sons, Inc.